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Aug 22 Define a Line using Vectors

Write the equation of a line containing the points (2,3,4) and (4,5,9).

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We can start by writing the "slope", which is just the vector between the points, which is $<2,2,5>$. It's important to note that this line is parallel to the vector $<2,2,5>$, so we're not done yet. We can say the line equals $(2, 3, 4) + t<2,2,5>$. Now, if we combine this into one vector, it's $ <2 + 2t, 3 + 2t, 4 + 5t> $. That's called the vector form. The parametric form is made by splitting the vector up into its separate parts.
$$\text{Vector form: }$$
$$r = (2,3,4) + t\cdot <2,2,5>$$
$$\text{Parametric form: }$$
$$x(t) = 2 + 2t $$
$$y(t) = 3 + 2t $$
$$z(t) = 4 + 5t $$

Write the equation of a line parallel to the previous line, but containing the point (17, 2, 3).